Noncausal Estimation for Discrete Gauss-Markov Random Fields

Abstract

In [1], it was shown that 2-D discrete Gauss-Markov random fields can be characterized in terms of a noncausal nearest-neighbor model (NNM) driven by locally correlated noise. This result is used here to obtain a simple solution of the smoothing problem for Gauss-Markov random fields. It is shown that the smoother has a nearest-neighbor structure of the same type as the original field, and that the smoothing error is itself a Gauss-Markov random field. Since the operator describing the smoother dynamics is positive and self-adjoint, the smoother can be implemented by using efficient iterative algorithms for elliptic PDEs.